Difference between revisions of "Extras"
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[[Category:Cosmic Microwave Background (CMB)|6]] | [[Category:Cosmic Microwave Background (CMB)|6]] | ||
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+ | <div id="cmb2"></div> | ||
+ | <div style="border: 1px solid #AAA; padding:5px;"> | ||
+ | === Problem 1 === | ||
+ | In year 1953 the article "Extended Universe and creation of Galactics" by G.A. Gamov was published.<br/> | ||
+ | In that paper Gamov took two numbers - the age of the world and average density of matter in the Universe - and basing on them he determined the third number: the temperature of the relic radiation (cosmic microwave background). Try to repeat the scientific feat of Gamov. | ||
+ | <div class="NavFrame collapsed"> | ||
+ | <div class="NavHead">solution</div> | ||
+ | <div style="width:100%;" class="NavContent"> | ||
+ | <p style="text-align: left;">Gamov considered the simplest variant of cosmological dynamics---the inertial regime of the expansion of Universe: \( R \propto t,\ v = const. | ||
+ | \) Density of homogeneous Universe, filled by non-relativistic matter, is | ||
+ | \[ | ||
+ | \rho _m = \frac{M}{{(4/3\pi )R^3 }} = \rho _m (t_0 )\left( | ||
+ | {\frac{{t_0 }}{t}} \right)^3. | ||
+ | \] | ||
+ | Gamov used the following numerical numbers for current age of Universe and the matter density respectively: | ||
+ | \( t_0 \simeq 3\cdot^10^9\mbox{years}, | ||
+ | \rho _m (t_0 ) \simeq 10^{-30} \mbox{\it g/cm}^3.\) Energy density in the radiation dominated Universe is | ||
+ | \[ | ||
+ | \rho _r = \frac{3}{{32\pi Gt^2 }}. | ||
+ | \] | ||
+ | Then Gamov used the condition $\rho _r (t^*) = \rho _m (t^*)$ to find the moment of time when radiation-dominated epoch changed to matter-dominated one: | ||
+ | \( | ||
+ | t^* \simeq 73\cdot10^6\ \mbox{years}\) Then he made use of the Stephan-Boltzmann law to determine the temperature in the joining point, i.e. at the time moment $t^*$ , \( T(t^*) = 320K.\) It is left only to use the condition $aT = const,$ in order to convert the temperature for the present time to obtain the following | ||
+ | \[ | ||
+ | T(t_0 ) = T\left( {t^*} \right){\frac{{t^*}}{{t_0 }}} \simeq 7K. | ||
+ | \] | ||
+ | |||
+ | "Is it possible to take two asymptotes, join them and determine the CMB temperature? The answer is, firstly, it is impossible, and secondly, Gamov did it already long ago." (A.D. Chernin)</p> | ||
+ | </div> | ||
+ | </div></div> |
Revision as of 21:24, 1 October 2012
Problem 1
In year 1953 the article "Extended Universe and creation of Galactics" by G.A. Gamov was published.
In that paper Gamov took two numbers - the age of the world and average density of matter in the Universe - and basing on them he determined the third number: the temperature of the relic radiation (cosmic microwave background). Try to repeat the scientific feat of Gamov.
Gamov considered the simplest variant of cosmological dynamics---the inertial regime of the expansion of Universe: \( R \propto t,\ v = const. \) Density of homogeneous Universe, filled by non-relativistic matter, is \[ \rho _m = \frac{M}[[:Template:(4/3\pi )R^3]] = \rho _m (t_0 )\left( {\frac[[:Template:T 0]]{t}} \right)^3. \] Gamov used the following numerical numbers for current age of Universe and the matter density respectively: \( t_0 \simeq 3\cdot^10^9\mbox{years}, \rho _m (t_0 ) \simeq 10^{-30} \mbox{\it g/cm}^3.\) Energy density in the radiation dominated Universe is \[ \rho _r = \frac{3}[[:Template:32\pi Gt^2]]. \] Then Gamov used the condition $\rho _r (t^*) = \rho _m (t^*)$ to find the moment of time when radiation-dominated epoch changed to matter-dominated one: \( t^* \simeq 73\cdot10^6\ \mbox{years}\) Then he made use of the Stephan-Boltzmann law to determine the temperature in the joining point, i.e. at the time moment $t^*$ , \( T(t^*) = 320K.\) It is left only to use the condition $aT = const,$ in order to convert the temperature for the present time to obtain the following \[ T(t_0 ) = T\left( {t^*} \right){\frac[[:Template:T^*]][[:Template:T 0]]} \simeq 7K. \] "Is it possible to take two asymptotes, join them and determine the CMB temperature? The answer is, firstly, it is impossible, and secondly, Gamov did it already long ago." (A.D. Chernin)